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  排序-01-堆排序
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      <h1 id="排序-01-堆排序"><a href="#排序-01-堆排序" class="headerlink" title="排序-01-堆排序"></a>排序-01-堆排序</h1><h2 id="前言"><a href="#前言" class="headerlink" title="前言"></a>前言</h2><ul>
<li>堆排序是利用<strong>堆</strong>这种数据结构而设计的一种排序算法，堆排序是一种<strong>选择排序，</strong>它的最坏，最好，平均时间复杂度均为O(nlogn)，它也是不稳定排序。首先简单了解下堆结构。</li>
</ul>
<h2 id="1-堆"><a href="#1-堆" class="headerlink" title="1. 堆"></a>1. 堆</h2><ul>
<li>堆是具有如下性质的完全二叉树<ul>
<li>每个节点的值都大于或者等于其左右孩子节点的值，成为大顶堆</li>
<li>每个节点的值都小于或者等于其左右孩子节点的值，成为小顶堆</li>
</ul>
</li>
</ul>
<p><img src="http://zhuuu-bucket.oss-cn-beijing.aliyuncs.com/img/20200901/210713650.png" alt="mark"></p>
<p>同时，将堆中节点按照层级进行编号，将这种逻辑结构映射到数组中就是如下的例子</p>
<p><img src="http://zhuuu-bucket.oss-cn-beijing.aliyuncs.com/img/20200901/210812092.png" alt="mark"></p>
<p>该数组从逻辑上讲就是一个堆结构，我们用简单的公式来描述一下堆的定义就是：</p>
<p><strong>大顶堆：arr[i] &gt;= arr[2i+1] &amp;&amp; arr[i] &gt;= arr[2i+2]</strong>  </p>
<p><strong>小顶堆：arr[i] &lt;= arr[2i+1] &amp;&amp; arr[i] &lt;= arr[2i+2]</strong>  </p>
<p>ok，了解了这些定义。接下来，我们来看看堆排序的基本思想及基本步骤：</p>
<a id="more"></a>



<h2 id="2-堆排序介绍"><a href="#2-堆排序介绍" class="headerlink" title="2. 堆排序介绍"></a>2. 堆排序介绍</h2><ul>
<li>堆排序(Heap Sort) 是指利用堆这种数据结构所涉及的一种排序算法。</li>
<li><strong>将n个无序序列构建成一个堆，根据升序降序需求选择大顶堆或小顶堆;</strong></li>
<li><strong>将堆顶元素与末尾元素交换，将最大元素”沉”到数组末端;</strong></li>
<li><strong>重新调整结构，使其满足堆定义，然后继续交换堆顶元素与当前末尾元素，反复执行调整+交换步骤，直到整个序列有序。</strong></li>
</ul>
<p><strong>步骤一 构造初始堆。将给定无序序列构造成一个大顶堆（一般升序采用大顶堆，降序采用小顶堆)。</strong></p>
<p><img src="http://zhuuu-bucket.oss-cn-beijing.aliyuncs.com/img/20200901/211132796.png" alt="mark"></p>
<p><strong>步骤二 将堆顶元素与末尾元素进行交换，使末尾元素最大。然后继续调整堆，再将堆顶元素与末尾元素交换，得到第二大元素。如此反复进行交换、重建、交换。</strong></p>
<h2 id="3-大顶堆实现堆排"><a href="#3-大顶堆实现堆排" class="headerlink" title="3. 大顶堆实现堆排"></a>3. 大顶堆实现堆排</h2><figure class="highlight java"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br></pre></td><td class="code"><pre><span class="line"> </span><br><span class="line"><span class="keyword">import</span> java.util.Arrays;</span><br><span class="line"><span class="comment">/**</span></span><br><span class="line"><span class="comment"> * </span></span><br><span class="line"><span class="comment"> * <span class="doctag">@author</span> Administrator</span></span><br><span class="line"><span class="comment"> *</span></span><br><span class="line"><span class="comment"> */</span></span><br><span class="line"><span class="keyword">public</span> <span class="class"><span class="keyword">class</span> <span class="title">HeapSort</span> </span>&#123;</span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">main</span><span class="params">(String []args)</span></span>&#123;</span><br><span class="line">        <span class="keyword">int</span> []arr = &#123;<span class="number">7</span>,<span class="number">6</span>,<span class="number">7</span>,<span class="number">11</span>,<span class="number">5</span>,<span class="number">12</span>,<span class="number">3</span>,<span class="number">0</span>,<span class="number">1</span>&#125;;</span><br><span class="line">        System.out.println(<span class="string">"排序前："</span>+Arrays.toString(arr));</span><br><span class="line">        sort(arr);</span><br><span class="line">        System.out.println(<span class="string">"排序前："</span>+Arrays.toString(arr));</span><br><span class="line">    &#125;</span><br><span class="line"> </span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">sort</span><span class="params">(<span class="keyword">int</span> []arr)</span></span>&#123;</span><br><span class="line">        <span class="comment">//1.构建大顶堆</span></span><br><span class="line">        <span class="keyword">for</span>(<span class="keyword">int</span> i=arr.length/<span class="number">2</span>-<span class="number">1</span>;i&gt;=<span class="number">0</span>;i--)&#123;</span><br><span class="line">            <span class="comment">//从第一个非叶子结点从下至上，从右至左调整结构</span></span><br><span class="line">            adjustHeap(arr,i,arr.length);</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="comment">//2.调整堆结构+交换堆顶元素与末尾元素</span></span><br><span class="line">        <span class="keyword">for</span>(<span class="keyword">int</span> j=arr.length-<span class="number">1</span>;j&gt;<span class="number">0</span>;j--)&#123;</span><br><span class="line">            swap(arr,<span class="number">0</span>,j);<span class="comment">//将堆顶元素与末尾元素进行交换</span></span><br><span class="line">            adjustHeap(arr,<span class="number">0</span>,j);<span class="comment">//重新对堆进行调整</span></span><br><span class="line">        &#125;</span><br><span class="line"> </span><br><span class="line">    &#125;</span><br><span class="line"> </span><br><span class="line">    <span class="comment">/**</span></span><br><span class="line"><span class="comment">     * 调整大顶堆（仅是调整过程，建立在大顶堆已构建的基础上）</span></span><br><span class="line"><span class="comment">     * <span class="doctag">@param</span> arr</span></span><br><span class="line"><span class="comment">     * <span class="doctag">@param</span> i</span></span><br><span class="line"><span class="comment">     * <span class="doctag">@param</span> length</span></span><br><span class="line"><span class="comment">     */</span></span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">adjustHeap</span><span class="params">(<span class="keyword">int</span> []arr,<span class="keyword">int</span> i,<span class="keyword">int</span> length)</span></span>&#123;</span><br><span class="line">        <span class="keyword">int</span> temp = arr[i];<span class="comment">//先取出当前元素i</span></span><br><span class="line">        <span class="keyword">for</span>(<span class="keyword">int</span> k=i*<span class="number">2</span>+<span class="number">1</span>;k&lt;length;k=k*<span class="number">2</span>+<span class="number">1</span>)&#123;<span class="comment">//从i结点的左子结点开始，也就是2i+1处开始</span></span><br><span class="line">            <span class="keyword">if</span>(k+<span class="number">1</span>&lt;length &amp;&amp; arr[k]&lt;arr[k+<span class="number">1</span>])&#123;<span class="comment">//如果左子结点小于右子结点，k指向右子结点</span></span><br><span class="line">                k++;</span><br><span class="line">            &#125;</span><br><span class="line">            <span class="keyword">if</span>(arr[k] &gt;temp)&#123;<span class="comment">//如果子节点大于父节点，将子节点值赋给父节点（不用进行交换）</span></span><br><span class="line">                arr[i] = arr[k];</span><br><span class="line">                i = k;</span><br><span class="line">            &#125;<span class="keyword">else</span>&#123;</span><br><span class="line">                <span class="keyword">break</span>;</span><br><span class="line">            &#125;</span><br><span class="line">        &#125;</span><br><span class="line">        arr[i] = temp;<span class="comment">//将temp值放到最终的位置</span></span><br><span class="line">    &#125;</span><br><span class="line"> </span><br><span class="line">    <span class="comment">/**</span></span><br><span class="line"><span class="comment">     * 交换元素</span></span><br><span class="line"><span class="comment">     * <span class="doctag">@param</span> arr</span></span><br><span class="line"><span class="comment">     * <span class="doctag">@param</span> a</span></span><br><span class="line"><span class="comment">     * <span class="doctag">@param</span> b</span></span><br><span class="line"><span class="comment">     */</span></span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">swap</span><span class="params">(<span class="keyword">int</span> []arr,<span class="keyword">int</span> a ,<span class="keyword">int</span> b)</span></span>&#123;</span><br><span class="line">        <span class="keyword">int</span> temp=arr[a];</span><br><span class="line">        arr[a] = arr[b];</span><br><span class="line">        arr[b] = temp;</span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>



<p><img src="http://zhuuu-bucket.oss-cn-beijing.aliyuncs.com/img/20200812-202023330.gif" alt="mark"></p>
<h2 id="4-小顶堆实现堆排以及TopK"><a href="#4-小顶堆实现堆排以及TopK" class="headerlink" title="4. 小顶堆实现堆排以及TopK"></a>4. 小顶堆实现堆排以及TopK</h2><p><strong>小顶堆实现 (堆排序 以及 topk)</strong></p>
<figure class="highlight java"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">public</span> <span class="class"><span class="keyword">class</span> <span class="title">HeapSort</span></span>&#123;</span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">heapSort</span><span class="params">(<span class="keyword">int</span>[] tree,<span class="keyword">int</span> n)</span> </span>&#123;</span><br><span class="line">        buildHeap(tree, n);<span class="comment">//第一步是将得到的数组构建成小顶堆</span></span><br><span class="line">        <span class="keyword">for</span>(<span class="keyword">int</span> i = n-<span class="number">1</span>;i&gt;=<span class="number">0</span>;i--) &#123;</span><br><span class="line">            swap(tree, i, <span class="number">0</span>);<span class="comment">//第一次构建完小顶堆之后，要进行第一个数和最后一个树的交换</span></span><br><span class="line">            <span class="comment">//交换完之后，最上面的数就不是最小数了，因此只需要对最上面的数，进行一个树的调整即可</span></span><br><span class="line">            <span class="comment">//所以，我们使用的时adjustTree而不是buildHeap</span></span><br><span class="line">            adjustTree(tree, i, <span class="number">0</span>);<span class="comment">//这里解释一下，这参数的含义：之所以将i当做数组的长度，</span></span><br><span class="line">            <span class="comment">//是因为我们将第一个数和最后一个数交换之后，就已经把最小的数放在了数组最后，进行</span></span><br><span class="line">            <span class="comment">//树调整的时候，就不需要管最后一个数字了。而0就是因为交换之后需要进行节点调节的那个节点</span></span><br><span class="line">            <span class="comment">//换到了第一个位置</span></span><br><span class="line">        &#125;</span><br><span class="line">    &#125;        </span><br><span class="line">    <span class="comment">/*</span></span><br><span class="line"><span class="comment">     * 这个函数写完之后，就可以将任意一个数组，构建成小顶堆了，构建完小顶堆之后，就要进行堆排序了</span></span><br><span class="line"><span class="comment">     */</span></span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">buildHeap</span><span class="params">(<span class="keyword">int</span>[] tree,<span class="keyword">int</span> n)</span> </span>&#123;</span><br><span class="line">        <span class="keyword">for</span>(<span class="keyword">int</span> i = (n-<span class="number">1</span>)/<span class="number">2</span>;i&gt;=<span class="number">0</span>;i--) &#123;<span class="comment">//i从最后一个子节点的父节点开始，所以i = (n-1)/2</span></span><br><span class="line">            adjustTree(tree, n, i);</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="comment">//利用adjustTree和swap两个函数，可以针对某一个父节点，进行调节。接下来，解决当整个树是</span></span><br><span class="line">    <span class="comment">//乱序的，将一个树构建成一个小顶堆。思路是这样的：从最后一个子节点的父节点开始调节，往上走。</span></span><br><span class="line">    <span class="comment">//不断重复，每往上一个父节点，父节点的下标就减一，可以将adjustTree和swap函数放进一个for循环</span></span><br><span class="line">    <span class="comment">//就是上面的for循环</span></span><br><span class="line">    <span class="comment">/*</span></span><br><span class="line"><span class="comment">     * 表示从某一个节点开始，调整一次树，使之成为堆，其中i表示某一个节点的下标</span></span><br><span class="line"><span class="comment">     */</span></span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">adjustTree</span><span class="params">(<span class="keyword">int</span>[] tree,<span class="keyword">int</span> n,<span class="keyword">int</span> i)</span> </span>&#123;</span><br><span class="line">        <span class="keyword">if</span>(i&gt;=n) &#123;<span class="comment">//这是递归头。</span></span><br><span class="line">            <span class="keyword">return</span>;</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="comment">//首先确定i节点的左右两个孩子的下标</span></span><br><span class="line">        <span class="keyword">int</span> c1 = <span class="number">2</span>*i+<span class="number">1</span>;</span><br><span class="line">        <span class="keyword">int</span> c2 = <span class="number">2</span>*i+<span class="number">2</span>;</span><br><span class="line">        <span class="comment">//接下来，在这三个值中，找出最小值</span></span><br><span class="line">        <span class="keyword">int</span> max = i;<span class="comment">//先假设最小值为这个父节点</span></span><br><span class="line">        <span class="keyword">if</span>(c1&lt;n &amp;&amp; tree[c1]&gt;tree[max]) &#123;<span class="comment">//要保证c1不会出界</span></span><br><span class="line">            max = c1;</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">if</span>(c2&lt;n &amp;&amp; tree[c2]&gt;tree[max]) &#123;<span class="comment">//保证c2不会出界  c2&lt;n</span></span><br><span class="line">            max = c2;</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="comment">//经过上面的条件判断，就可以将最小值的下标保存到max中了，如果最小值max就是i，也就是</span></span><br><span class="line">        <span class="comment">//父节点最小，就不用调整，但是如果父节点不是最小，就要进行交换了</span></span><br><span class="line">        <span class="keyword">if</span>(max!=i) &#123;</span><br><span class="line">            swap(tree,max,i);</span><br><span class="line">            adjustTree(tree,n,max);<span class="comment">//交换之后，将父节点下放一级，就有可能会破坏下一层结构，</span></span><br><span class="line">            <span class="comment">//所以，递归调用adjustTree.使用递归之后，就要添加递归头了</span></span><br><span class="line">        &#125;              </span><br><span class="line">    &#125;</span><br><span class="line">    <span class="function"><span class="keyword">private</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">swap</span><span class="params">(<span class="keyword">int</span>[] tree, <span class="keyword">int</span> i, <span class="keyword">int</span> j)</span> </span>&#123;</span><br><span class="line">        <span class="keyword">int</span> temp = tree[i];</span><br><span class="line">        tree[i] = tree[j];</span><br><span class="line">        tree[j] = temp;</span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>



<figure class="highlight java"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">public</span> <span class="class"><span class="keyword">class</span> <span class="title">TopK</span> </span>&#123;</span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">main</span><span class="params">(String[] args)</span> </span>&#123;</span><br><span class="line"></span><br><span class="line">        <span class="keyword">int</span>[] data = <span class="keyword">new</span> <span class="keyword">int</span>[]&#123;<span class="number">1</span>, <span class="number">3</span>, <span class="number">4</span>, <span class="number">2</span>, <span class="number">8</span>, <span class="number">9</span>, <span class="number">5</span>, <span class="number">6</span>, <span class="number">7</span>, <span class="number">32</span>, <span class="number">56</span>, <span class="number">23</span>, <span class="number">87</span>, <span class="number">32</span>&#125;;<span class="comment">//原始数据</span></span><br><span class="line"></span><br><span class="line">        <span class="keyword">int</span>[] topk = topK(data, <span class="number">5</span>);<span class="comment">//调用topK方法，返回前k大的数组，返回的数组并不是有序的，而是一个小顶堆，如果想返回一个有序的，可以调用上面HeapSort类中的heapSort方法</span></span><br><span class="line"></span><br><span class="line">        <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i &lt; topk.length; i++) &#123;<span class="comment">//循环输出小顶堆</span></span><br><span class="line">            System.out.println(topk[i]);</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">int</span>[] topK(<span class="keyword">int</span>[] data, <span class="keyword">int</span> k) &#123;</span><br><span class="line">        <span class="keyword">int</span>[] topk = <span class="keyword">new</span> <span class="keyword">int</span>[k];<span class="comment">//根据传进来的K创建长度为k的数组</span></span><br><span class="line"></span><br><span class="line">        <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i &lt; k; i++) &#123;</span><br><span class="line">            topk[i] = data[i];<span class="comment">//先将源数据的前k个的数赋值给topK数组</span></span><br><span class="line">        &#125;</span><br><span class="line"></span><br><span class="line">        HeapSort.buildHeap(topk, k);<span class="comment">//对这个topK数组进行一次最小堆的构建。</span></span><br><span class="line"></span><br><span class="line">        <span class="keyword">for</span> (<span class="keyword">int</span> i = k; i &lt; data.length; i++) &#123;<span class="comment">//从源数据的第K个数开始循环，如果循环的数比堆顶元素还小，直接pass，</span></span><br><span class="line">            <span class="comment">// 如果比堆顶元素要大，就将此数放在堆顶，同时进行一次以它为起始点的树的调整。</span></span><br><span class="line">            <span class="keyword">int</span> temp = data[i];</span><br><span class="line">            <span class="keyword">if</span> (topk[<span class="number">0</span>] &lt; temp) &#123;</span><br><span class="line">                topk[<span class="number">0</span>] = temp;</span><br><span class="line">                HeapSort.adjustTree(topk, k, <span class="number">0</span>);</span><br><span class="line">            &#125;</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">return</span> topk;</span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>



<p>再简单总结下堆排序的基本思路：</p>
<p>　　<strong>a.将无序序列构建成一个堆，根据升序降序需求选择大顶堆或小顶堆;</strong></p>
<p>　　<strong>b.将堆顶元素与末尾元素交换，将最大元素”沉”到数组末端;</strong></p>
<p>　　<strong>c.重新调整结构，使其满足堆定义，然后继续交换堆顶元素与当前末尾元素，反复执行调整+交换步骤，直到整个序列有序。</strong></p>

      
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